r H = r H = j!E+ E = j!E r E = ;j!H p Ex(z t) = jc1j

W.C.Chew
ECE 350 Lecture Notes
14. Skin Depth and Plane Wave in a Lossy Medium.
We learn earlier that in a lossy medium, J = E, and from
r H = @E
@t
+ J = @E
@t
Using phasor technique, we can convert the above to
where
+ E: (1)
r H = j! E + E = j! E (2)
is the complex permittivity. Furthermore, using that
and that r H =0,r E =0,we can show that
= ; j ! (3)
r E = ;j! H (4)
r 2 E = ;! 2 E (5)
r 2 H = ;! 2 H: (6)
[Refer to x 4 for details]. If we assume that E = ^xE x(z), then, we can show
that
d 2
dz 2 E x(z) ; 2 E x(z) =0 (7)
where
= j! p = + j : (7a)
The general solution to (7) is of the form
If we assume that c 2 =0,we have only
E x(z) =c 1e ; z + c 2e z : (8)
E x(z) =c 1e ; z : (9)
We can convert the above into a real time quantity using phasor techniques,
or
E x(z t) =jc 1j

where we have assumed that c 1 = jc 1j e j 1 . Hence, we see that Ex(z t) is
a wave that propagates to the right with velocity v = ! and attenuation
constant . We can nd from equation (7a), and
r r
= + j = j! ; j
!
= j! 1 ; j
!
: (11)
The rst term on the RHS of (1) is the displacement current term, while the
second term is the conduction current term. From (2), we see that the ratio
! is the ratio of the conduction current to the displacement current in a lossy
medium. is also known as the loss tangent of a lossy medium.
!
(i) When 1, the loss tangent is small, and the conduction current com-
!
pared to the displacement current is small. The medium behaves more
like a dielectric medium. In this case, we can use binomial expansions to
approximate (11) to obtain
= j! p
1 ; j 1
=
2 !
1
r
p
+ j! (12)
2
where
= 1
2
r = ! p : (13)
(ii) When 1, the loss tangent is large because there is more conduc-
!
tion current than displacement current in the medium. In this case, the
medium is conductive. According to equation (11), when 1, we
!
have
r
= j! ;j =
! p r
!
j! =(1+j) : (14)
2
Hence
= =
If we substitute = = 1 into (10), we have
r !
2 = 1 : (15)
Ex(z t) =jc1j e ;z
cos !t ; z + 1 : (16)
2

Ex(z,t)
1
0.8
0.6
0.4
2.
0.2
2.5
0
−0.2
−0.4
−0.6
−0.8
1.5
t=0.
1.
.5
Ex(z,t), t=0.,0.5,1.0,1.5,2.0,2.5
omega=1, delta=0.2, phi=−0.25pi
−1
0 0.1 0.2 0.3 0.4 0.5
z
0.6 0.7 0.8 0.9 1
This signal attenuates to e ;1 of its original strength at z = . Hence
is also known as the penetration depth or the skin depth of a conductive
medium. For other media, the penetration is 1 , but for a conductive medium,
it is
r
2
=
!
r
1
=
f
: (17)
This skin depth decreases with increasing frequencies and increasing conductivities.
(iii) When 1, it is a general lossy medium, and we have to resort to
!
complex arithmatics to nd and .
If we square (11), we have
or
2 ; 2 +2j = ;! 2 ( ; j ! ) (18)
2 ; 2 = ;! 2
(19a)
2 = ! : (19b)
Squaring (19a) and adding the square of (19b) to it, we have
or
( 2 ; 2 ) 2 +(2 ) 2 =( 2 + 2 ) 2 = ! 4 2 2 + ! 2 2 2 (20)
Combining with (19a), we deduce that
2 + 2 = ! p ! 2 2 + 2 : (21)
2 = 1
2 (! p ! 2 2 + 2 ; ! 2
3
) (22a)

2 = 1
Notice that when =0, =0.
2 (! p ! 2 2 + 2 + ! 2
4
) (22b)